Proofs with transformations

Problem

Rotations preserve lengths and angles. AB‾≅CD‾{\overline{AB}}\cong{\overline{CD}}​AB​​​≅​CD​​​, so we know ϕ=θ\purple\phi=\pink\thetaϕ=θ.

B

OA‾≅OD‾{\overline{OA}}\cong{\overline{OD}}​OA​​​≅​OD​​​ and OB‾≅OC‾{\overline{OB}}\cong{\overline{OC}}​OB​​​≅​OC​​​. This means △AOC≅△DOB{\triangle{AOC}}\cong{\triangle{DOB}}△AOC≅△DOB. Since the triangles are congruent, we know ϕ=θ\purple\phi=\pink\thetaϕ=θ.

C

If and are each rotated 180∘180^\circ180​∘​​180, degree about point OOOO, they must map to and respectively. If two rays are rotated by the same amount, the angle between them will not change, so ϕ\purple\phiϕ must be equal to θ\pink\thetaθstart color pink, theta, end color pink.